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Scalar function on space
Scalar function on space
Value at every node
[ E.g. put multiple scalar functions to specify “coordinate” ]
Tangent space
Tangent space
Equivalence class of geodesics emanating from a point
Vectors
Vectors
[[ Basic question: is a vector a single geodesic, or a linear combination of geodesics? ]]
A vector is a geodesic joining two points [on a manifold, it’s OK just to give the points]
Possible vectors from a given point are the possible geodesics, going a certain distance along the geodesic
< Also, could have a vector that is a linear combination of two edges >
Multiplication by scalars
Multiplication by scalars
Go further along the vector [multiply by integers]
Adding vectors
Adding vectors
Not so obvious, because of parallel transport
[ Go down one geodesic, then from the end point go to another geodesic => non-commutative when there is curvature ]
[ In making a vector space ... normally integer dimensional ]
Vector function on space
Vector function on space
To every point in the space, assign a particular geodesic away from that point as the “vector value” at that point
Vector field
Vector field
Derivation of scalar function [ e.g. directional derivative at each point ]
[ Non-commutative geometry: the “scalar values” do not form a commutative ring; but then one does derivatives anyway ]
< Alternative: at every point, pick an edge >
At every point, associate a number with every edge at that point [ i.e. think of the edges at a point as the basis elements for a vector ]
(When this limits to a manifold, there have to be relations between the numbers on different edges)
[[ With definition of vector as linear combination, derivation is taking differences along each edge ]]
Tensor field
Tensor field
Rank p tensor field: at every point, pick p edges [allowing repeats]
Each collection of p edges is a like basis element in the tensor Tuples[edges, p]
Tensor
Tensor
Basis is tuples of vectors at a given point
<|1w1,3w2,5w3|>(*vector*)
<|{1,3}w1,{2,4}w2,...|>(*2-tensor*)
Dual vectors
Dual vectors
Assume a vector is defined by a number associated with each outgoing edge
[[Linear function from vectors to scalars]] f[new vector] : like a projection onto the existing vector
< cotangent space >
Volume form
Volume form
All the outgoing edges around a given point
(Dual to) p-Form
(Dual to) p-Form
Basis is all collections of all-but-p of the outgoing edges
Tensor product
Tensor product
I.e. outer product, where one forms all combinations of the keys
Wedge product
Wedge product
u⋀v : take all the edges associated with u and concatenate them with v
When the resulting component is {1,3}, the weight is the negative of when it’s {3,1}
Totally antisymmetric tensor [?]
Totally antisymmetric tensor [?]
<|{even}1,{odd}-1,XXX|>
<|{1,2,3,4,5}1|>
Rotation groups
Rotation groups